A Chernoff-type bound for local hamiltonian systems on quantum lattice

Abstract

We consider the problem of determining the energy distribution of a product state, with respect to a local hamiltonian in a quantum lattice. For a one dimensional lattice with \(n\) sites and a \(2\)-local hamiltonian defined on it, we show that the overlap of a product state (with average energy \(f\)) with the eigenspace of energy \(g\) is less that \(e^{-(g-f)^2/112n}\), as long as \(|g-f|>102n^{1/2}\). We prove a weaker result for states satisfying exponential decay of correlation: given a quantum state with correlation length \(\sigma\) and average energy \(f\), its overlap with eigenspace of energy \(g\) is less than \(\mathcal{O}(1)e^{-|g-f|/\sqrt{54n\sigma}}\), as long as \(|g-f|\geq \mathcal{O}(1)(n/\sigma)^{1/2}\).